I would like to sum over a partial interval:
$$ \sum_{t = -k}^{0}\beta_{k}, $$
where $\beta$ could be any constant (e.g., $\beta = 5$). I want to express an additive sum from $-k$ to $0$. The negative constant in the lower limit is an arbitrary naming convention.
Question:
(1) Is it proper to call the $t$ a "variable" and $-k$ a constant? Should $\beta$ be $t$-subscripted?
- The real "variable" is $t$, which is a dummy variable.
- It is important to show this explicitly as some academic papers like to express outcomes of a linear equation before some event. I use the negative subscripts to explicitly denote the periods before exposure to some intervention.
(2) Could someone review what I have tried:
Assuming $\beta = 5$,
$$ \sum_{t = -k}^{0}5 = 5(0 - (-k) + 1) = 5(0 + k + 1) = 5k + 5, $$
and if $k=3$ (which is an arbitrary subscript), then the solution is:
$$ \sum_{t = -3}^{0}5 = 5(0 - (-3) + 1) = 5(0 + 3 + 1) = 5(4) = 20. $$
Any thoughts?
Your work is mathematically correct.
Your answers for both examples are correct and clearly explained.
Someone may ask why do you use the index $k$ in $\beta _k$ if it is a constant.